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{{Infobox Philosopher | |
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region = Classical Greek philosophy | |
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era = Ancient philosophy | |
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color = #B0C4DE | |
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image_name = Domenico-Fetti Archimedes 1620.jpg | |
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image_caption = ''Archimedes Thoughtful'' by ] (1620) | |
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<!-- Information --> |
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name = Archimedes of Syracuse (Greek: Άρχιμήδης) | |
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birth = ''c''. ] (], ]) | |
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death = ''c''. ] (Syracuse) | |
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school_tradition = ]<br> ] | |
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main_interests = ], ], ], ], ] | |
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influences = | |
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influenced = | |
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notable_ideas = ], ]s, <br>] | |
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}} |
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'''Archimedes''' (]: {{polytonic|Άρχιμήδης}} ''c''. ] – ''c''. ]) was an ancient ] ], ], ], ], and ]. He is widely regarded as the most important ] in ]. |
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==Biography== |
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Archimedes was born ''c''. ] in the seaport city of ], which was then a colony of ]. The date of his birth is based on an assertion by the ] historian ] that he lived for seventy-five years. In '']'' Archimedes gives his father's name as Phidias, an ] about whom nothing is known. ] wrote that Archimedes was related to King ], the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work is lost, causing many details of his life to remain obscure.<ref name="mactutor">{{cite web | author=J. J. O'Connor, E. F. Robertson | url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Archimedes.html | title = Archimedes of Syracuse | publisher = University of St Andrews | accessdate = 2007-01-02 }}</ref> Archimedes was educated in ], ], which was the greatest center of learning in the world at the time. There he became friends with ] and ]. Some of the mathematical works of Archimedes were written in the form of letters to Eratosthenes, who was the chief ] in Alexandria. |
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Archimedes died ''c''. ] during the ], when ] forces under General ] captured the city of Syracuse after a two year long ]. According to the popular account, Archimedes was busy contemplating a mathematical drawing in the sand. He was interrupted by a Roman soldier and replied impatiently: "Don't disturb my circles" (μή μου τούς κύκλους τάραττε). The soldier was enraged by this, and killed Archimedes with his sword. The quote is often given in ] as "''Noli turbare circulos meos''" but there is no direct evidence that Archimedes ever uttered these words. General Marcellus was reportedly angered by the death of Archimedes, as he had ordered him not to be harmed.<ref>{{cite web | url = http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html | title = Death of Archimedes: Sources | publisher = Courant Institute of Mathematical Sciences | accessdate = 2007-01-02 }}</ref> |
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The tomb of Archimedes had a carving of his favorite mathematical diagram, which was a ] inside a ] of the same height and diameter. Archimedes had proved that the volume and surface area of the sphere would be two thirds that of the cylinder. In ], 137 years after the death of Archimedes, the ] ] ] visited the tomb in ] which had become overgrown with scrub. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.<ref>{{cite web | url = https://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html | title = Tomb of Archimedes: Sources | publisher = Courant Institute of Mathematical Sciences | accessdate = 2007-01-02 }}</ref><ref>{{cite web | url = https://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html | title = Archimedes on Spheres and Cylinders | publisher = MathPages | accessdate = 2007-01-02 }}</ref> |
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Some of the classical accounts of the life of Archimedes were written long after his death and contain ]s of questionable authenticity. The version of the siege of Syracuse given by ] is likely to have been compiled from first hand evidence, and was used as a source by later historians including ] and ]. <ref>https://www.math.nyu.edu/~crorres/Archimedes/Siege/Polybius.html</ref> |
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], himself frequently called the most influential mathematician of all time, stated that Archimedes was one of the three epoch-making mathematicians, with the others being ] and ]. |
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There is a ] on the ] named ] in his honor, along with a lunar mountain range, the ]. |
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==Discoveries and inventions== |
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Archimedes is regarded as the first ], and he was the key contributor to this field prior to ] and ]. The most famous anecdote told about his work is how he discovered the principle of ]. According to ], a new crown in the shape of a ] had been made for King Hieron, and Archimedes was asked to determine whether it was of solid ], or whether ] had been added by a dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down in order to measure its ]. While taking a bath, he noticed that the level of the water rose as he got in. He realized that this effect could be used to determine the volume of the crown, and therefore its ] after weighing it. The density of the crown would be lower if cheaper and lighter metals had been added. He then took to the streets naked, being so elated with his discovery that he forgot to dress, crying "]!" ("I have found it!").<ref>http://hyperphysics.phy-astr.gsu.edu/Hbase/pbuoy.html</ref><ref>http://www.dctech.com/eureka/archimedes/crown/</ref> This discovery is known in the field of ] as ], which states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.<ref>http://www.physics.weber.edu/carroll/Archimedes/principle.htm</ref> |
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] |
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Another invention bearing his name is the ]. This was a machine with a revolving screw shaped blade, and was used to drain ships and transfer water from a low-lying body of water into irrigation canals. Versions of the Archimedes screw are still in use today in developing countries. |
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While Archimedes did not invent the ], he gave the first rigorous explanation of the principles involved, which are the transmission of force through a ] and moving the effort applied through a greater distance than the object to be moved. His ''Law of the Lever'' states: ''Magnitudes are in equilibrium at distances reciprocally proportional to their weights''. His work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth."<ref>https://www.math.nyu.edu/~crorres/Archimedes/Lever/LeverIntro.html</ref> Plutarch describes how Archimedes designed ] ] systems, allowing sailors to use the principle of ] to lift objects that would otherwise have been too heavy to move. |
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A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer ] describes how King Hieron II commissioned Archimedes to design a huge ship, the ''Syracusia'', which could be used for luxury travel, carrying supplies and as a naval warship. The ''Syracusia'' is said to have been the largest ship built in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and contained garden decorations, a ] and a ] dedicated to the goddess ]. Since a ship of this size would leak a considerable amount of water through the hull, the ] was purportedly developed in order to remove the bilge water.<ref>http://www.mlahanas.de/Greeks/Syracusia.htm</ref> |
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] |
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During the ], Archimedes is said to have repelled an attack by ] forces by using a "]" to focus sunlight on the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes death ray", has been the subject of ongoing debate about its credibility since the ]. ] rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes. It has been suggested that a large array of highly polished ] shields acting as ]s could have been employed to focus sunlight on to a ship, utilizing the principle of the ]. In October 2005 a group of students from the ] carried out an experiment with 127 one foot square mirror tiles, focused on a mocked-up wooden ship at a range of around 100 feet. Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. Nevertheless, it was concluded that the weapon was a feasible device under these conditions. The MIT group repeated the experiment for the ] show '']'', using a wooden fishing boat in ] as the target. Again some charring occurred, along with a small amount of flame. When ''Mythbusters'' broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "myth" due to the length of time and ideal weather conditions required for combustion to occur. Critics of the MIT experiments have argued that the moisture content of the wood needs to be taken into consideration. However, the ] of wood is 300 degrees ] (572 degrees ]), and this is hotter than the maximum temperature produced by domestic ]s. <ref>http://science.howstuffworks.com/wildfire.htm</ref> In real life a ] (]) armed with flaming bolts would have been a more dangerous weapon, while the effects of the "Archimedes death ray" might have been limited to confusing or temporarily blinding people on board the ship.<ref>http://web.mit.edu/2.009/www//experiments/deathray/10_ArchimedesResult.html</ref><ref>http://web.mit.edu/2.009/www//experiments/deathray/10_Mythbusters.html</ref><ref>http://www.mlahanas.de/Greeks/Mirrors.htm</ref> |
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The ] is another weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal ] was suspended. When the claw was dropped on to an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. As with the "Archimedes death ray" there have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled ''Superweapons of the Ancient World'' built a version of the claw and concluded that it was a workable device. No contemporary drawings of the Claw of Archimedes exist, although the weapon may have been similar in design to a ].<ref>https://www.math.nyu.edu/~crorres/Archimedes/Claw/illustrations.html</ref> |
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Archimedes has also been credited with improving the power and accuracy of the ], and with inventing the ] during the ]. The odometer is said to have been a ] with a ] mechanism that dropped a ball into a container after each mile traveled. <ref>http://www.mlahanas.de/Greeks/ArchimedesGears.htm</ref> |
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] wrote that after the capture of ], General ] took two mechanical devices back to ] that were used as aids in ]. He credits ] and ] with constructing these devices. The motions of the ], ] and five ] were shown by one device, and it was demonstrated to Cicero some 150 years later by a man named Gallus. Cicero described the event as follows: |
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:''Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione.'' - When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze as in the sky itself, from which also in the sky the Sun's globe became that same eclipse, and the Moon came then to that position which was shadow the Earth, when the Sun was in line.<ref>http://www.thelatinlibrary.com/cicero/repub1.shtml#21</ref> |
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The device described by Cicero is a ] or ]. ] stated that Archimedes had written a manuscript (now lost) on the construction of these devices entitled '']''. Recent research in this area has been focused on the ], another device from ] that was probably designed for the same purpose. Constructing devices of this kind would have required a sophisticated knowledge of ]. This was once thought to have been beyond the range of the ] available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient ].<ref>https://www.math.nyu.edu/~crorres/Archimedes/Sphere/SphereIntro.html</ref><ref>http://news.bbc.co.uk/1/hi/sci/tech/6191462.stm</ref> |
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==Mathematics== |
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Although he is often regarded as a designer of mechanical devices, Archimedes also made important contributions to the field of ]. ] wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.” |
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] to calculate the value of ]]] |
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Some of his mathematical proofs involve the use of ] in a way that is similar to modern ]. By assuming a propostion to be true and showing that this would lead to a ], Archimedes was able to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the ], and he employed it to calculate the value of ] (Pi). He did this by drawing a larger ] outside a ], and a smaller polygon inside the circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). This was a remarkable achievement, since the ancient Greek number system was unwieldy and used letters rather than the symbols used today. He also proved that the ] of a circle was equal to π multiplied by the ] of the ] of the circle. |
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He used the method of exhaustion to show that the value of the ] of 3 lay between 265/153 (approximately 1.732) and 1351/780 (approximately 1.7320512). The modern value is ~1.7320508076, making this a very accurate estimate. |
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Another noted mathematical work by Archimedes is '']''. In this work he set out to calculate the number of grains of ] that the ] could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelon (Gelon II, son of ]), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the ]. This was the ancient Greek word for ], based on the Greek word for uncountable, ''murious''. The word myriad was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8{{e|63}} in modern notation.<ref>http://physics.weber.edu/carroll/Archimedes/sand.htm</ref> |
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<div style="float:right;padding:5px;text-align:center">]<br></div> |
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In the field of ], Archimedes proved that the area enclosed by a ] and a straight line is 4/3 multiplied by the area of a ] with equal base and height (see illustration on right). |
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He expressed the solution to the problem as a ] that summed to ] with the ] 1/4: |
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:<math> \sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3} \; . </math> |
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If the first term in this series is the area of the triangle in the illustration then the second is the sum of the areas of two triangles whose bases are the two smaller ]s in the illustration, and so on. Archimedes also gave a different proof of nearly the same ] by using ]. |
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It has been suggested that ] for calculating the area of a ] from the length of its sides was known to Archimedes. However, the first reliable reference to this formula occurs in the work of ] in the ] ]. <ref>http://mathworld.wolfram.com/HeronsFormula.html</ref> |
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==The Archimedes Palimpsest== |
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{{main|Archimedes Palimpsest}} |
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The work of Archimedes was not as widely recognized in ] as that of ], and a number of his writings are believed to have been lost when the ] was burned at various periods in its history. Some of the writings of Archimedes survived through ] and ] translations made during the ], and it is these documents that provide much of the modern knowledge of his treatises. The most important document containing his work is the ]. A ] is a document written on ] that has been re-used by scraping off the ink of an older text and writing new text in its place. In 1906, the ] professor ] realized that a goatskin parchment containing prayers written in the ] also carried an older work written in the ], which he identified as previously unknown copies of works by Archimedes. The parchment spent many years in a ] library in ] before being sold to a private collector, and reappeared at an auction at ] in ] in October 1998 where it was sold to an anonymous buyer for $2 million. The Archimedes Palimpsest contains seven treatises, including the only surviving copy of ''On Floating Bodies'' in the original Greek. Most importantly, it contains the only known source of the ''Method of Mechanical Theorems'' and ''Stomachion'', which had previously been thought lost. The Archimedes Palimpsest is now stored at the ] in ], ], where it has been subjected to a range of modern tests including the use of ] and ] ] to read the overwritten text.<ref>http://news.bbc.co.uk/1/hi/sci/tech/5235894.stm</ref> |
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The treatises contained in the Archimedes Palimpsest are: ''On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems'' and ''Stomachion''. |
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==Writings by Archimedes== |
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] |
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* ''On the Equilibrium of Planes'' (2 volumes) |
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:This treatise explains the ''Law of the Lever'', and uses it to calculate the ] and ] of various geometric figures including ], ]s, and ]s. |
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* ''On Spirals'' |
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:In this treatise he defines what is now called an ]. This is the first example of a ] (a curve traced by a moving ]) considered by a Greek mathematician. |
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* ''On the Sphere and the Cylinder'' |
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:In this treatise Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a ] ] of the same height and ]. He proves that the sphere will have exactly two thirds of the volume and area of the cylinder. A carving of this proof was used on the tomb of Archimedes. |
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* ''On Conoids and Spheroids'' |
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:In this treatise Archimedes calculates the areas and volumes of ] of ]s, spheres, and paraboloids. |
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* ''On Floating Bodies'' (2 volumes) |
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:In the first part of this treatise, Archimedes spells out the law of ] of fluids, and proves that water will adopt a spherical form around a center of gravity. This was probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. His fluids are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape. |
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:In the second part, he calculates the equilibrium positions of ] of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that ]s float. |
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* ''The Quadrature of the Parabola'' |
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: In this treatise he proves that the area enclosed by a ] and a straight line is 4/3 multiplied by the area of a ] with equal base and height. He achieves this by calculating the value of a ] that sums to infinity with the ] 1/4. (See the diagram in ]). |
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] ] from 1983. He is demonstrating the quadrature of the ], while the ] demonstrates the principle of ].]] |
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* ''Stomachion'' |
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:This is a Greek puzzle similar to a ], and the treatise describing it was found in more complete form in the ]. Archimedes calculates the areas of the various pieces. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a ]. This is possibly the first use of ] to solve a problem. |
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* '']'' |
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:Archimedes wrote a letter to the scholars in the ], who had apparently questioned the importance of his work. He challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous ]s. There is a more difficult version of the problem where some of the answers are required to be ]. This version of the problem was first solved by a ] in 1965, and the answer is a very large number, approximately 7.760271{{e|206544}}.<ref>http://mathworld.wolfram.com/ArchimedesCattleProblem.html</ref> |
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* '']'' |
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:In this treatise, Archimedes counts the number of grains of sand that will fit inside the ]. This book mentions ]' theory of the ] (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the ], Archimedes concludes that the number of grains of sand required to fill the universe is 8{{e|63}} in modern notation. The introductory letter contains the information that the father of Archimedes was an ] named Phidias. |
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* '']'' |
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:This treatise was thought lost until the discovery of the ] in 1906. In this work Archimedes uses ]s, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the ] to derive the results. <!-- This particular work is found in what is called the ]. --> |
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==See also== |
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* ] |
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* ] |
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* ] |
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* ] |
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* ] |
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* ] |
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==References== |
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<div class="references-small"><references/></div> |
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==Books== |
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* ], ''Archimedes'', 1987, Princeton University Press, Princeton, ISBN 0-691-08421-1. Republished translation of the 1938 study of Archimedes and his works by an historian of science. |
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* ], ''Works of Archimedes'', Dover Publications, ISBN 0-486-42084-1. Complete works of Archimedes in English. |
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== External links == |
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{{wikiquote}} |
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* - ] Radio 4 discussion from '']'', broadcast 25 January 2007 (requires ]). |
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* at ] |
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* by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas. |
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* |
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* {{MacTutor Biography|id=Archimedes }} |
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* web pages at the ]. |
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* |
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* |
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* points out that in reality Archimedes may well have used a more subtle method than the one in the classic version of the story. |
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* Translated by ]. |
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* Translated by ]. |
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* by ]. |
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*{{gutenberg author|id=Archimedes|name=Archimedes}} |
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